1. Response to Intervention www.interventioncentral.org Foundations of Math Skills & RTI Interventions Jim Wright www.interventioncentral.org

2. Response to Intervention www.interventioncentral.org 2 ‘Elbow Group’ Activity: What are common student math concerns in your school? In your ‘elbow groups’: Discuss the most common math problems that you encounter in your school(s). At what grade level do you typically encounter these problems? Be prepared to share your discussion points with the larger group.

3. Response to Intervention www.interventioncentral.org 3 Profile of Students with Math Difficulties (Kroesbergen & Van Luit, 2003) [ Although the group of students with difficulties in learning math is very heterogeneous], in general, these students have memory deficits leading to difficulties in the acquisition and remembering of math knowledge. Moreover, they often show inadequate use of strategies for solving math tasks, caused by problems with the acquisition and the application of both cognitive and metacognitive strategies. Because of these problems, they also show deficits in generalization and transfer of learned knowledge to new and unknown tasks. Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and Special Education, 24, 97-114..

4. Response to Intervention www.interventioncentral.org 4 Three General Levels of Math Skill Development (Kroesbergen & Van Luit, 2003) As students move from lower to higher grades, they move through levels of acquisition of math skills, to include: Number sense Basic math operations (i.e., addition, subtraction, multiplication, division) Problem-solving skills: “The solution of both verbal and nonverbal problems through the application of previously acquired information” ( Kroesbergen & Van Luit, 2003, p. 98 ) Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and Special Education, 24, 97-114..

5. Response to Intervention www.interventioncentral.org 5 What is ‘Number Sense’? (Clarke & Shinn, 2004) “… the ability to understand the meaning of numbers and define different relationships among numbers. Children with number sense can recognize the relative size of numbers, use referents for measuring objects and events, and think and work with numbers in a flexible manner that treats numbers as a sensible system. ” p. 236 Source: Clarke, B., & Shinn, M. (2004). A preliminary investigation into the identification and development of early mathematics curriculum-based measurement. School Psychology Review, 33, 234–248.

6. Response to Intervention www.interventioncentral.org 6 What Are Stages of ‘Number Sense’? (Berch, 2005, p. 336) 1.Innate Number Sense. Children appear to possess ‘hard- wired’ ability (neurological ‘foundation structures’) to acquire number sense. Children’s innate capabilities appear also to be to ‘represent general amounts’, not specific quantities. This innate number sense seems to be characterized by skills at estimation (‘approximate numerical judgments’) and a counting system that can be described loosely as ‘1, 2, 3, 4, … a lot’. 2.Acquired Number Sense. Young students learn through indirect and direct instruction to count specific objects beyond four and to internalize a number line as a mental representation of those precise number values. Source: Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38, 333-339...

7. Response to Intervention www.interventioncentral.org 7 Benefits of Automaticity of ‘Arithmetic Combinations’ (: (Gersten, Jordan, & Flojo, 2005) There is a strong correlation between poor retrieval of arithmetic combinations (‘math facts’) and global math delays Automatic recall of arithmetic combinations frees up student ‘cognitive capacity’ to allow for understanding of higher-level problem-solving By internalizing numbers as mental constructs, students can manipulate those numbers in their head, allowing for the intuitive understanding of arithmetic properties, such as associative property and commutative property Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293-304.

8. Response to Intervention www.interventioncentral.org 8 Associative Property “within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed” Example: –(2+3)+5=10 – 2+(3+5)=10 Source: Associativity. Wikipedia. Retrieved September 5, 2007, from http://en.wikipedia.org/wiki/Associative

9. Response to Intervention www.interventioncentral.org 9 Commutative Property “the ability to change the order of something without changing the end result.” Example: – 2+3+5=10 – 2+5+3=10 Source: Associativity. Wikipedia. Retrieved September 5, 2007, from http://en.wikipedia.org/wiki/Commutative

10. Response to Intervention www.interventioncentral.org 10 How much is 3 + 8?: Strategies to Solve… Least efficient strategy: Count out and group 3 objects; count out and group 8 objects; count all objects: + =11 More efficient strategy: Begin at the number 3 and ‘count up’ 8 more digits (often using fingers for counting): 3 + 8 More efficient strategy: Begin at the number 8 (larger number) and ‘count up’ 3 more digits: 8 + 3 Most efficient strategy: ‘3 + 8’ arithmetic combination is stored in memory and automatically retrieved: Answer = 11 Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293-304.

11. Response to Intervention www.interventioncentral.org 11 Internal Numberline As students internalize the numberline, they are better able to perform ‘mental arithmetic’ (the manipulation of numbers and math operations in their head). 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 2 + 4 = 6 28 ÷ 4 = 7 9 – 7 = 2 3 X 7 = 21

12. Response to Intervention www.interventioncentral.org 12 Math Skills: Importance of Fluency in Basic Math Operations “[A key step in math education is] to learn the four basic mathematical operations (i.e., addition, subtraction, multiplication, and division). Knowledge of these operations and a capacity to perform mental arithmetic play an important role in the development of children’s later math skills. Most children with math learning difficulties are unable to master the four basic operations before leaving elementary school and, thus, need special attention to acquire the skills. A … category of interventions is therefore aimed at the acquisition and automatization of basic math skills.” Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and Special Education, 24, 97-114.

13. Response to Intervention www.interventioncentral.org 13 Think-aloud and the Internal Numberline in Action: What is 37 multiplied by 46? Well, let’s see. First, I know that 30 times 46 would be like multiplying ‘46 times 10’ three times in a row. That would be, um, 460 times 3. Three times zero is zero [ones place value], 6 times 3 is 18 [tens place value]…carry a one and add it to ‘4 times 3’ to give you 13. So 460 times 3 is 1380. And that takes care of ’30 times 46’. Now I have to solve for ‘7 times 46’. Hmmm…7 times 40 would be 280…I know that because ‘7 times 4 is 28…just add another zero. I can then add 1380 and 280—and that would be 1660. I knew that because 1380 plus 300 is 1680 and then I just subtracted 20. What’s left? Um…7 times 6. That would be 42. So 1660 and 42 would be, uh…[subvocally] 1670, 1680, 1690, 1700…and two. [Aloud] The answer is 1702.

14. Response to Intervention www.interventioncentral.org 14 Big Ideas: Learn Unit (Heward, 1996) The three essential elements of effective student learning include: 1.Academic Opportunity to Respond. The student is presented with a meaningful opportunity to respond to an academic task. A question posed by the teacher, a math word problem, and a spelling item on an educational computer ‘Word Gobbler’ game could all be considered academic opportunities to respond. 2.Active Student Response. The student answers the item, solves the problem presented, or completes the academic task. Answering the teacher’s question, computing the answer to a math word problem (and showing all work), and typing in the correct spelling of an item when playing an educational computer game are all examples of active student responding. 3.Performance Feedback. The student receives timely feedback about whether his or her response is correct—often with praise and encouragement. A teacher exclaiming ‘Right! Good job!’ when a student gives an response in class, a student using an answer key to check her answer to a math word problem, and a computer message that says ‘Congratulations! You get 2 points for correctly spelling this word!” are all examples of performance feedback. Source: Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student response during group instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi (Eds.), Behavior analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.

15. Response to Intervention www.interventioncentral.org 15 Math Intervention: Tier I or II: Elementary & Secondary: Self-Administered Arithmetic Combination Drills With Performance Self-Monitoring & Incentives 1.The student is given a math computation worksheet of a specific problem type, along with an answer key [Academic Opportunity to Respond]. 2.The student consults his or her performance chart and notes previous performance. The student is encouraged to try to ‘beat’ his or her most recent score. 3.The student is given a pre-selected amount of time (e.g., 5 minutes) to complete as many problems as possible. The student sets a timer and works on the computation sheet until the timer rings. [Active Student Responding] 4.The student checks his or her work, giving credit for each correct digit (digit of correct value appearing in the correct place-position in the answer). [Performance Feedback] 5.The student records the day’s score of TOTAL number of correct digits on his or her personal performance chart. 6.The student receives praise or a reward if he or she exceeds the most recently posted number of correct digits. Application of ‘Learn Unit’ framework from : Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student response during group instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi (Eds.), Behavior analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.

16. Response to Intervention www.interventioncentral.org 16 Self-Administered Arithmetic Combination Drills: Examples of Student Worksheet and Answer Key Worksheets created using Math Worksheet Generator. Available online at: http://www.interventioncentral.org/htmdocs/tools/mathprobe/addsing.php

17. Response to Intervention www.interventioncentral.org 17 Self-Administered Arithmetic Combination Drills… No Reward Reward Given No Reward Reward Given

18. Response to Intervention www.interventioncentral.org 18 Math Intervention: Tier I or II: Elementary & Middle School: Cover Copy Compare The student is given a math worksheet with 10 number problems and answers on the left side of the page. For each problem, the student: –Studies the correctly completed problem on the left side of the page. –Covers the problem with an index card. –Copies the problem from memory on the right side of the page. –Solves the problem. –Uncovers the correct model problem to check his or her work. –If the student’s problem was done incorrectly, the student repeats the process until correct. Source: Skinner, C. H., Turco, T. L., Beatty, K. L., & Rasavage, C. (1989). Cover, copy, and compare: A method for increasing multiplication performance. School Psychology Review, 18, 412-420.

19. Response to Intervention www.interventioncentral.org 19 Math Intervention: Tier I: High School: Peer Guided Pause Students are trained to work in pairs. At one or more appropriate review points in a math lecture, the instructor directs students to pair up to work together for 4 minutes. During each Peer Guided Pause, students are given a worksheet that contains one or more correctly completed word or number problems illustrating the math concept(s) covered in the lecture. The sheet also contains several additional, similar problems that pairs of students work cooperatively to complete, along with an answer key. Student pairs are reminded to (a) monitor their understanding of the lesson concepts; (b) review the correctly math model problem; (c) work cooperatively on the additional problems, and (d) check their answers. The teacher can direct student pairs to write their names on the practice sheets and collect them to monitor student understanding. Source: Hawkins, J., & Brady, M. P. (1994). The effects of independent and peer guided practice during instructional pauses on the academic performance of students with mild handicaps. Education & Treatment of Children, 17 (1), 1-28.

20. Response to Intervention www.interventioncentral.org 20 RTI: Individual Case Study: Math Computation Jared is a fourth-grade student. His teacher, Mrs. Rogers, became concerned because Jared is much slower in completing math computation problems than are his classmates.

21. Response to Intervention www.interventioncentral.org 21 Tier 1: Math Interventions for Jared Jared’s school uses the Everyday Math curriculum (McGraw Hill/University of Chicago). In addition to the basic curriculum the series contains intervention exercises for students who need additional practice or remediation. The instructor, Mrs. Rogers, works with a small group of children in her room—including Jared—having them complete these practice exercises to boost their math computation fluency.

22. Response to Intervention www.interventioncentral.org 22 Tier 2: Standard Protocol (Group): Math Interventions for Jared Jared did not make sufficient progress in his Tier 1 intervention. So his teacher referred the student to the RTI Intervention Team. The team and teacher decided that Jared would be placed on the school’s educational math software, AMATH Building Blocks, a ‘self-paced, individualized mathematics tutorial covering the math traditionally taught in grades K-4’. Jared worked on the software in 20-minute daily sessions to increase computation fluency in basic multiplication problems.

23. Response to Intervention www.interventioncentral.org 23 Tier 2: Math Interventions for Jared (Cont.) During this group-based Tier 2 intervention, Jared was assessed using Curriculum- Based Measurement (CBM) Math probes. The goal was to bring Jared up to at least 40 correct digits per 2 minutes.

24. Response to Intervention www.interventioncentral.org 24 Tier 2: Math Interventions for Jared (Cont.) Progress-monitoring worksheets were created using the Math Computation Probe Generator on Intervention Central ( www.interventioncentral.org ). Example of Math Computation Probe: Answer Key

25. Response to Intervention www.interventioncentral.org 25 Tier 2: Phase 1: Math Interventions for Jared: Progress-Monitoring

26. Response to Intervention www.interventioncentral.org 26 Tier 2: Individualized Plan: Math Interventions for Jared Progress-monitoring data showed that Jared did not make expected progress in the first phase of his Tier 2 intervention. So the RTI Intervention Team met again on the student. The team and teacher noted that Jared counted on his fingers when completing multiplication problems. This greatly slowed down his computation fluency. The team decided to use a research- based strategy, Cover-Copy-Compare, to increase Jared’s computation speed and eliminate his dependence on finger- counting. During this individualized intervention, Jared continued to be assessed using Curriculum-Based Measurement (CBM) Math probes. The goal was to bring Jared up to at least 40 correct digits per 2 minutes.

27. Response to Intervention www.interventioncentral.org 27 Cover-Copy-Compare: Math Computational Fluency-Building Intervention The student is given sheet with correctly completed math problems in left column and index card. For each problem, the student: –studies the model –covers the model with index card –copies the problem from memory –solves the problem –uncovers the correctly completed model to check answer Source: Skinner, C.H., Turco, T.L., Beatty, K.L., & Rasavage, C. (1989). Cover, copy, and compare: A method for increasing multiplication performance. School Psychology Review, 18, 412-420.

28. Response to Intervention www.interventioncentral.org 28 Tier 2: Phase 2: Math Interventions for Jared: Progress-Monitoring

29. Response to Intervention www.interventioncentral.org 29 Tier 2: Math Interventions for Jared Cover-Copy-Compare Intervention: Outcome The progress-monitoring data showed that Jared was well on track to meet his computation goal. At the RTI Team follow-up meeting, the team and teacher agreed to continue the fluency-building intervention for at least 3 more weeks. It was also noted that Jared no longer relied on finger-counting when completing number problems, a good sign that he had overcome an obstacle to math computation.

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32. Response to Intervention www.interventioncentral.org The application to create CBM Early Math Fluency probes online http://www.interventioncentral.org/php/numberfly/ numberfly.php

33. Response to Intervention www.interventioncentral.org 33 Examples of Early Math Fluency (Number Sense) CBM Probes Quantity Discrimination Missing Number Number Identification Sources: Clarke, B., & Shinn, M. (2004). A preliminary investigation into the identification and development of early mathematics curriculum-based measurement. School Psychology Review, 33, 234–248. Chard, D. J., Clarke, B., Baker, S., Otterstedt, J., Braun, D., & Katz, R. (2005). Using measures of number sense to screen for difficulties in mathematics: Preliminary findings. Assessment For Effective Intervention, 30(2), 3-14

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35. Response to Intervention www.interventioncentral.org 35 Potential ‘Blockers’ of Higher-Level Math Problem-Solving: A Sampler Limited reading skills Failure to master--or develop automaticity in– basic math operations Lack of knowledge of specialized math vocabulary (e.g., ‘quotient’) Lack of familiarity with the specialized use of known words (e.g., ‘product’) Inability to interpret specialized math symbols (e.g., ‘4 < 2’) Difficulty ‘extracting’ underlying math operations from word/story problems or identifying and ignoring extraneous information included in word/story problems

36. Response to Intervention www.interventioncentral.org CBM: Math Computation

37. Response to Intervention www.interventioncentral.org 37 CBM Math Computation Probes: Preparation

38. Response to Intervention www.interventioncentral.org 38 CBM Math Computation Sample Goals Addition: Add two one-digit numbers: sums to 18 Addition: Add 3-digit to 3-digit with regrouping from ones column only Subtract 1-digit from 2-digit with no regroupingSubtraction: Subtract 1-digit from 2-digit with no regrouping Subtract 2-digit from 3-digit with regrouping from ones and tens columnsSubtraction: Subtract 2-digit from 3-digit with regrouping from ones and tens columns Multiply 2-digit by 2-digit-no regroupingMultiplication: Multiply 2-digit by 2-digit-no regrouping Multiply 2-digit by 2-digit with regroupingMultiplication: Multiply 2-digit by 2-digit with regrouping

39. Response to Intervention www.interventioncentral.org 39 CBM Math Computation Assessment: Preparation Select either single-skill or multiple-skill math probe format. Create student math computation worksheet (including enough problems to keep most students busy for 2 minutes) Create answer key

40. Response to Intervention www.interventioncentral.org 40 CBM Math Computation Assessment: Preparation Advantage of single-skill probes: –Can yield a more ‘pure’ measure of student’s computational fluency on a particular problem type

41. Response to Intervention www.interventioncentral.org 41 CBM Math Computation Assessment: Preparation Advantage of multiple-skill probes: –Allow examiner to gauge student’s adaptability between problem types (e.g., distinguishing operation signs for addition, multiplication problems) –Useful for including previously learned computation problems to ensure that students retain knowledge.

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43. Response to Intervention www.interventioncentral.org 43 CBM Math Computation Probes: Administration

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45. Response to Intervention www.interventioncentral.org 45 CBM Math Computation Probes: Scoring

46. Response to Intervention www.interventioncentral.org 46 CBM Math Computation Assessment: Scoring Unlike more traditional methods for scoring math computation problems, CBM gives the student credit for each correct digit in the answer. This approach to scoring is more sensitive to short-term student gains and acknowledges the child’s partial competencies in math.

47. Response to Intervention www.interventioncentral.org Math Computation: Scoring Example 12 CDs

48. Response to Intervention www.interventioncentral.org Math Computation: Scoring Placeholders Are Counted Numbers Above Line Are Not Counted

49. Response to Intervention www.interventioncentral.org 49 CBM Math Computation Activity: Score the number of correct digits on your math probe.

50. Response to Intervention www.interventioncentral.org 50 Trainer Question: What objections or concerns might teachers have about using CBM math computation probes? How would you address these concerns?